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Zbl 1012.65052
Hernández, M.A.
The Newton method for operators with Hölder continuous first derivative.
(English)
[J] J. Optimization Theory Appl. 109, No.3, 631-648 (2001). ISSN 0022-3239; ISSN 1573-2878/e

This paper is devoted to the Newton method for solving the equation $$F(x)= 0\tag 1$$ starting at $x_0$ and defined by $$x_{n+1}= x_n- F'(x_n)^{-1} F(x_n)$$ with function $F'(x)$ satisfying $$\|F'(x)- F'(y)\|\le \kappa\|x- y\|^p,\quad p\in [0,1].$$ Based on this study the author obtains results on the existence and uniqueness of the solution of a nonlinear Hammerstein integral equation of the second kind.
[S.I.Piskarev (Moskva)]
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
47J25 Methods for solving nonlinear operator equations (general)
45G10 Nonsingular nonlinear integral equations
65R20 Integral equations (numerical methods)

Keywords: Newton method; semilocal convergence theorem; Hammerstein integral equation

Cited in: Zbl 1204.65058

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